2024-06-30-bateni24a.md

title	section	abstract	layout	series	publisher	issn	id	month	tex_title	firstpage	lastpage	page	order	cycles	bibtex_author	author	date	address	container-title	volume	genre	issued	pdf	extras
Metric Clustering and MST with Strong and Weak Distance Oracles	Original Papers	We study optimization problems in a metric space $(\mathcal{X},d)$ where we can compute distances in two ways: via a “strong” oracle that returns exact distances $d(x,y)$, and a “weak” oracle that returns distances $\tilde{d}(x,y)$ which may be arbitrarily corrupted with some probability. This model captures the increasingly common trade-off between employing both an expensive similarity model (e.g. a large-scale embedding model), and a less accurate but cheaper model. Hence, the goal is to make as few queries to the strong oracle as possible. We consider both “point queries”, where the strong oracle is queried on a set of points $S \subset \cX $ and returns $d(x,y)$ for all $x,y \in S$, and “edge queries” where it is queried for individual distances $d(x,y)$. Our main contributions are optimal algorithms and lower bounds for clustering and Minimum Spanning Tree (MST) in this model. For $k$-centers, $k$-median, and $k$-means, we give constant factor approximation algorithms with only $\tilde{O}(k)$ strong oracle point queries, and prove that $\Omega(k)$ queries are required for any bounded approximation. For edge queries, our upper and lower bounds are both $\tilde{\Theta}(k^2)$. Surprisingly, for the MST problem we give a $O(\sqrt{\log n})$ approximation algorithm using no strong oracle queries at all, and we prove a matching $\Omega(\sqrt{\log n})$ lower bound which holds even if $\Tilde{\Omega}(n)$ strong oracle point queries are allowed. Furthermore, we empirically evaluate our algorithms, and show that their quality is comparable to that of the baseline algorithms that are given all true distances, but while querying the strong oracle on only a small fraction ($<1%$) of points.	inproceedings	Proceedings of Machine Learning Research	PMLR	2640-3498	bateni24a	0	Metric Clustering and MST with Strong and Weak Distance Oracles	498	550	498-550	498	false	Bateni, MohammadHossein and Dharangutte, Prathamesh and Jayaram, Rajesh and Wang, Chen	given family MohammadHossein Bateni given family Prathamesh Dharangutte given family Rajesh Jayaram given family Chen Wang	2024-06-30		Proceedings of Thirty Seventh Conference on Learning Theory	247	inproceedings	date-parts 2024 6 30	https://proceedings.mlr.press/v247/bateni24a/bateni24a.pdf